GMSK spread modulation

ABSTRACT

Spread spectrum GMSK signals are presented. Transmission may involve obtaining a sequence of data symbols, obtaining a spread spectrum code comprising a sequence of spread spectrum chips, generating a sequence of pre-modulation chips by combining the sequence of data symbols with the spread spectrum chips, wherein for each data symbol, at least one of the pre-modulation chips is generated by taking into account at least the data symbol and at least one of the spread spectrum chips, performing Gaussian Minimum Shift Keying (GMSK) modulation using the sequence of pre-modulation chips to produce a spread spectrum GMSK signal, and transmitting the spread spectrum GMSK signal.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional Application No.60/515,864; filed Oct. 29, 2003. The 60/515,864 application isincorporated herein by reference.

BACKGROUND OF THE INVENTION

The present invention relates generally to the field of digitalcommunications and more specifically to modulation and spread spectrumtechnologies. Spread spectrum techniques have proven to be extremelyeffective in building multiple access systems, combating or suppressingjamming and interference, transmitting signals at low power, andachieving message privacy from undesired listeners. Spread spectrumtechnology is characterized by a signal occupying an allocated bandwidththat is much wider than the data rate at which information iscommunicated. While many of the benefits and fundamental principals ofspread spectrum techniques are well-known, actual design of spreadspectrum systems can vary widely and face significant challenges interms of performance, cost, and other considerations.

Spread spectrum systems commonly involve Phase Shift Keying (PSK)modulation as the modulation technique used in transmitting spreadspectrum signals. PSK modulation refers to a general category ofmodulation schemes that may includes Binary Phase Shift Keying (BPSK),Quadrature Phase Shift Keying (QPSK), and others. PSK modulation schemesare often associated with “spill-over” of signal energy from theallocated bandwidth into adjacent bandwidths. This effect may be reducedby incorporating various envelope, waveform shaping, and/or othertechniques. However, the need to improve spectral efficiency ofmodulated signals remains a fundamental problem in spread spectrumsystems. This is especially true in limited bandwidth environments suchas wireless and satellite systems, where neighboring signals packedclosely together in the frequency spectrum may experience significantinterference from one another due to unwanted “spill-over” of eachsignal outside of its allocated bandwidth.

Another important consideration in the design of spread spectrum systemsis susceptibility to phase noise. In traditional multidimensional QPSKspread spectrum systems, the two dimensional data signals correspondingto in-phase (I) and quadrature (Q) components of the QPSK signal areeach spread independently by separate spreading codes. Such independentspreading of the I and Q components means the combined complex signaldoes not necessarily represent each data symbol using antipodalsignaling. This yields a modulation scheme characterized by degradedphase noise sensitivity as compared to antipodal modulation schemes.

U.S. Pat. No. 5,020,075 presents a “chip insertion” technique thatutilizes a spread spectrum pseudo-random (PN) sequence in providinginput to a minimum-shift-keying (MSK) modulator. In this patent, the PNsequence of chips is periodically interrupted with chips that representsymbols from a differentially encoded stream of data symbols. Theresulting pre-modulation chip stream is presented to aminimum-shift-keying (MSK) modulator. To achieve Gaussianminimum-shift-keying (GMSK) modulation, the technique replaces the MSKmodulator with a Gaussian filtered MSK modulator. However, this “chipinsertion” technique is prone to significant performance degradations.For example, in “Performance of DS/GMSK/PSK Modem Using Four-PhaseCorrelator”, Yano, et. al, IEEE Spread Spectrum Techniques andApplications Proceedings, 1996., pgs. 249–253, the performance of a QPSKchip insertion system is shown to be about 2 dB away from theoreticalperformance bounds.

Improvements to current design that address these and other shortcomingswould undoubtedly advance spread spectrum technology toward its fullpotential as a powerful communications methodology.

BRIEF SUMMARY OF THE INVENTION

The invention presents methods, apparatuses, and systems fortransmitting and receiving spread spectrum Gaussian Minimum Shift Keying(GMSK) signals. Transmission may involve obtaining a sequence of datasymbols, obtaining a spread spectrum code comprising a sequence ofspread spectrum chips, generating a sequence of pre-modulation chips bycombining the sequence of data symbols with the spread spectrum chips,wherein for each data symbol, at least one of the pre-modulation chipsis generated by taking into account at least the data symbol and atleast one of the spread spectrum chips, performing Gaussian MinimumShift Keying (GMSK) modulation using the sequence of pre-modulationchips to produce a spread spectrum GMSK signal, and transmitting thespread spectrum GMSK signal. The spread spectrum GMSK signal may adoptantipodal signaling to represent each data symbol. As an example, foreach data symbol, if the data symbol takes on a first binary value, acorresponding portion of the spread spectrum GMSK signal exhibits aselect waveform, and if the data symbol takes on a second binary value,the corresponding portion of the spread spectrum GMSK signal exhibits aphase-inverted version of the select waveform.

Each data symbol may correspond to K spread spectrum chips and Kpre-modulation chips, wherein K is a positive integer. A leading one ofthe K pre-modulation chips corresponding to a current data symbol may begenerated by taking into account (i) an immediately preceding datasymbol, (ii) a final one of the K spread spectrum chips corresponding tothe immediately preceding data symbol, (iii) the current data symbol,and (iv) a leading one of the K spread spectrum chips corresponding tothe current data symbol, whereas remaining ones of the K pre-modulationchips corresponding to the current data symbol may each be generated bytaking into account (i) an immediately preceding one of the K spreadspectrum chips corresponding to the current data symbol, and (ii) acurrent one of the K spread spectrum chips corresponding to the currentdata symbol. According to one embodiment, the spread spectrum codeutilized is a pseudo-noise (PN) code.

Reception may involve receiving a spread spectrum GMSK signalcorresponding to a sequence of data symbols and a spread spectrum code,processing the received spread spectrum GMSK signal using at least onematched filter to produce at least one filtered signal, correlating theat least one filtered signal with the spread spectrum code correspondingto the spread spectrum GMSK signal to produce at least one correlationoutput, and evaluating the correlation output to estimate the sequenceof data symbols.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a basic block diagram of a system for transmitting a GaussianMinimum Shift Keying (GMSK) modulation spread spectrum signal, inaccordance with one embodiment of the present invention;

FIG. 2 illustrates the pulse functions for Minimum Shift Keying (MSK)and the PAM expansion functions h₀(t), h₁(t), and h₂(t) for GMSK;

FIG. 3 is a table showing an example the mapping of data symbols d_(n)and spread spectrum code b_(k) into a pre-modulation chip sequenceα_(k), in order to produce Laurent expansion PAM symbols a_(0,k) thatare antipodal with respect to each data symbol d_(n);

FIG. 4 is a basic block diagram of a demodulator structure for receivinga GMSK modulation spread spectrum signal, in accordance with oneembodiment of the present invention;

FIG. 5 is a plot of computer simulation results for channel BERperformance versus Es/No;

FIG. 6 is a plot of the calculated loss versus different values of theGMSK BT product;

FIG. 7 illustrates the calculated power spectral density (PSK) ofvarious modulated signals, including offset QPSK, QPSK, MSK, and GMSKsignals of different BT products;

FIG. 8 is a plot showing the constant β tabulated for different valuesof the BT product; and

FIG. 9 is a table showing values of a figure of merit (FOM) for variousbandwidth definitions and BT products.

DETAILED DESCRIPTION OF THE INVENTION

Introduction

FIG. 1 is a basic block diagram of a system 100 for transmitting aspread spectrum Gaussian Minimum Shift Keying (GMSK) signal, inaccordance with one embodiment of the present invention. As shown in thefigure, a data symbol source block 104 provides a sequence of datasymbols d_(n), and a pseudo-random or pseudo-noise (PN) generator block102 provides spread spectrum code b_(k) to a mapping block 106. Thespread spectrum code b_(k) is also referred to as a sequence of spreadspectrum chips. The mapping block 106 utilizes the data symbols d_(n)and spread spectrum code b_(k) to produce a sequence of pre-modulation“chips” α_(k), which is provided to a GMSK modulator block 108. The GMSKmodulator block 108 performs GMSK modulation on the “chip” sequenceα_(k) to produce a GMSK waveform for transmission.

According to the present embodiment, the mapping block 106 combines datasymbols dn and spectrum code bk in such a way that the impression of thedata symbols on the GMSK waveform produced by the GMSK modulation block108 results in near antipodal signaling (BPSK). This provides a waveformcharacteristics of GMSK with data symbol performance and phase errorsensitivity similar to that of BPSK. In addition, GMSK modulation allowsspectral characteristics and Multiple Access Interference (MAI)performance to be easily traded off by adjustment of the GMSK parameterreferred to as bandwidth-bit duration product (BT), or BT product,associated with the GMSK modulator block 108. This allows the GMSKwaveform to be designed to meet the specific objectives of the system.

While a PN generator block 102 is shown in this figure, other codessuitable for spread spectrum applications may also be used in accordancewith the present invention. Further, data symbols d_(n) may broadlyrefer to any sequence of symbols representing data. For example, datasymbols d_(n) may represent information symbols that have not been errorcorrection encoded. Alternatively, data symbols d_(n) may representerror correction encoded symbols, such as Forward Error Correction (FEC)encoded symbols.

GMSK Description

According to the present embodiment, the input to the GMSK modulatorblock 108 is the “chip” sequence, α_(k). Here, the chips α_(k) take onbinary values of +1 or −1. For modulator synthesis, GMSK may bedescribed as a Continuous Phase Modulation (CPM) technique. The GMSKwaveform may be defined as:

$\begin{matrix}{{s(t)} = {{A\;{\cos\left( {{2\pi\; f} + {2\;\pi\; h{\sum\limits_{k = 0}^{m}{\alpha_{k}{q\left( {t - {kT}_{c}} \right)}}}}}\mspace{11mu} \right)}\mspace{11mu}{mT}_{c}} \leq t \leq {\left( {m + 1} \right)T_{c}}}} & (1)\end{matrix}$where f is the channel center frequency plus any requested frequencyoffset, T_(c) is the chip duration (1/chip rate), h=0.5, and

$\begin{matrix}\begin{matrix}{{q(t)} = {\int_{- \infty}^{t}{{g(x)}{\mathbb{d}x}}}} \\{{g(t)} = {\frac{1}{2T_{c}}\left\lbrack {{Q\left( {\sigma\; t_{1}} \right)} - {Q\left( {\sigma\; t_{2}} \right)}} \right\rbrack}} \\{t_{1} = {\frac{t}{T_{c}} - \left( \frac{L + 1}{2} \right)}} \\{t_{2} = {\frac{t}{T_{c}} - \left( \frac{L - 1}{2} \right)}} \\{\sigma = \frac{2\pi\;{BT}_{c}}{\sqrt{\ln(2)}}}\end{matrix} & (2)\end{matrix}$the function Q(t) is the standard Q function and is defined by:

$\begin{matrix}{{Q(t)} = {\frac{1}{\sqrt{2\pi}}{\overset{\infty}{\int\limits_{t}}{{\mathbb{e}}^{{- x^{2}}/2}{\mathbb{d}x}}}}} & (3)\end{matrix}$

Here, the product BT_(c) is used to correspond to the chip durationT_(c). The product BT_(c) serves as a parameter that can be varied totrade off better spectral performance (lower BT_(c)) vs. betterdetection efficiency (higher BT_(c)). To simplify implementation, it canbe roughly assumed for purposes of calculation that the frequency pulse,g(t) is only non-zero for a duration of about L chips (0≦t≦LT_(c)).Similarly, the phase pulse q(t) is essentially 0 for t<0 and essentiallyequal to ½ for t>LT_(c). As example, with BT_(c)=0.25, the value of L is4. For BT_(c)=0.20, the value of L should be increased to 5.

The description of q(t) in equation (2) is convenient for modulatorrealizations using a Frequency Modulation technique (such as a VCO orNCO). The integral in the description of q(t) above can be computed toget a direct expression for q(t):

$\begin{matrix}{{q(t)} = {\frac{1}{2}\begin{bmatrix}{1 + {t_{1}{Q\left( {\sigma\; t_{1}} \right)}} - {t_{2}Q\left( {\sigma\; t_{2}} \right)} -} \\{\frac{1}{\sigma\sqrt{2\pi}}\left( {{\mathbb{e}}^{{- \frac{1}{2}}\sigma^{2}t_{1}^{2}} - {\mathbb{e}}^{{- \frac{1}{2}}\sigma^{2}t_{2}^{2}}} \right)}\end{bmatrix}}} & (4)\end{matrix}$

This form is convenient for a realization using Phase Modulationsynthesis techniques.

Laurent Expansion

To determine the mapping structure that gives the spread spectrum GMSKsignal a BPSK nature to it, one may examine the Laurent PAMrepresentation, described in “Exact and Approximate Construction ofDigital Phase Modulations by Superposition of Amplitude ModulatedPulses”. P. A. Laurent, IEEE Transactions Communications, vol. COM-34,1986. Specifically, the baseband CPM signal consisting of K chips can berepresented as the sum of 2L−1 PAM signals:

$\begin{matrix}{{s(t)} = {{\mathbb{e}}^{j\;{\theta{(t)}}} = {\sum\limits_{l = 0}^{2^{L - 1} - 1}{\sum\limits_{k = 0}^{K - 1}{a_{l,k}{h_{l}\left( {t - {kT}_{c}} \right)}}}}}} & (5)\end{matrix}$

For many values of the BT_(c) product, the main pulse (corresponding tol=0) carries most of the signal energy. In this case, the only relevantpulse shapes and PAM data sequences are:

$\begin{matrix}{{h_{0}(t)} = {\frac{1}{\sqrt{T_{c}}}{\prod\limits_{i = 1}^{L}\;{c\left( {t - {iT}_{c}} \right)}}}} & (6) \\{a_{0,k} = {j^{({\sum\limits_{i = 0}^{k}\alpha_{i}})} = {a_{0,{k - 1}}j^{\alpha_{k}}}}} & (7)\end{matrix}$

The basic function c(t) is defined by:

$\begin{matrix}{{\left. \left. {{c(t)} = {\sin\left\lbrack {\frac{\pi}{2}\left( {1 - {2{q\left(  \right.}t}} \right.} \right)}} \right) \right\rbrack - {LT}_{c}} \leq t \leq {LT}_{c}} & (8)\end{matrix}$

Expressions for the other pulse functions and data symbols can be foundin the appendix of “Simple Coherent receivers for Partial responseContinuous Phase Modulation”, G. K. Kaleh, IEEE Journal on SelectedAreas in Communications, Vol 7, No 9, December 1989. From the recursionformula in equation (7), the PAM symbols are imaginary for n even andreal for n odd.

FIG. 2 illustrates the pulse functions, h_(l)(t), for Minimum ShiftKeying (MSK) and GMSK. For MSK,

${q(t)} = {\frac{1}{2}\frac{t}{T_{c}}}$for 0<t<T_(c), =0 for t<0, and =½ for t>T_(c) and L=1. The functionh₀(t) has a duration of 2T_(c) and is shown in FIG. 2. The result is thestandard offset QPSK view of MSK with sinusoidal pulse shaping and datamapping given by equation (7). Since the PAM data sequence a_(0,k)alternates between real and imaginary values, little or no intersymbolinterference is expected.

For GMSK with a BT_(c) of 0.25, an acceptable value of L is 4. Theduration of the main Laurent pulse is 5T_(c). The pulse shapes for theh₀(t), h₁(t), and h₂(t) Laurent pulses shown in the figure above. Onlythe first two pulses are significant. Also, since the duration of themain pulse is larger than 2T_(c), this waveform will produce intersymbolinterference.

Symbol Mapping

The objective for the mapping function is to combine the data symbolsd_(n) with the PN sequence b_(k) to produce a pre-modulation chipsequence α_(k) such that the Laurent PAM expansion results in antipodalsignaling for the main PAM pulse, h₀(t). In particular, it is desiredthat all the values of a_(0,k) to be inverted during data symbol n, butonly during data symbol n, if d_(n) is inverted. An additional objectiveis to have a simple relationship between the data values for the mainLaurent pulse (a_(0,k)) and the spread spectrum chips (b_(k)). Theseobjectives can be accomplished for the K chips within symbol interval nby using the following mapping:α_(m)=α_(k+(n−1)K)=(−1)^(m)({tilde over (b)} ⁻¹ {tilde over (d)}_(n−1))·({tilde over (b)} ₀ {tilde over (d)} _(n))k=0  (9)α_(m)=α_(k+(n−1)K)=(−1)^(m) {tilde over (b)} _(k−1) {tilde over (b)}_(k) k>0where {tilde over (b)}_(k)=2b_(k)−1 is the translation of the logicvalues of 0 and 1 to numerical values of −1 and 1 respectively. Likewisefor {tilde over (d)}_(n)=2d_(n)−1. The term {tilde over (b)}⁻¹ is thelast PN chip associated with data symbol d_(n−1) (the previous symbol).For the first symbol in the packet (d₀), the previous symbol, d⁻¹, andthe last PN chip of the previous symbol, b⁻¹, are both initialized to 1.The variable m is a chip counter that starts at 0 at the beginning oftransmission and does not get reset at the end of any symbol. Thevariable k is the chip count within a symbol, ranging from 0 to K−1. Inanother embodiment of the present invention, the alternate inversionperformed by the (−1)^(m) term can be equivalently implemented in thereceiver. In other words, the (−1)^(m) term in Equation 9 may be omittedin generating the pre-modulation chips at the transmitter end. Insteadthe (−1)^(m) term may be incorporated into the receiver structure.

FIG. 3 is a table showing an example of such mapping of data symbolsd_(n) and spread spectrum code b_(k) into a pre-modulation chip sequenceα_(k), in order to produce Laurent expansion PAM symbols a_(0,k) thatare antipodal with respect to each data symbol d_(n). For purposes ofillustration, K=4 chips per symbol is shown. The table illustrates themapping of the sequence of data symbols 0-1-1 using a particular PNsequence, then illustrates the mapping of the sequence of data symbols1-0-1 using the same PN sequence. As shown, the Laurent expansion PAMsymbols exhibit antipodal signaling with respect to each data symbol.The first symbol of the two sequences are different, so theircorresponding PAM expansions {−j, 1, −j, 1} and {+j,−1,+j, 1} areopposite. The second symbol of both sequences are also different, againyielding the two opposite PAM expansions {−j, 1, −j, 1} and {+j, −1, +j,−1}. Finally, the third symbol of both sequences are the same, yieldingthe identical PAM expansion {+j, 1, −j, −1 }. The simple relationshipbetween the PN chips and the I and Q values of the PAM symbols is alsoshown in the table. In particular, the 4 PN chips for the first datasymbol are {1,0,1,1 } and the four I and Q values for the PAM symbolsare {−1, 1, −1, −1}. In all cases, the I and Q values for the PAMsymbols are just the exclusive OR of the PN chip and the data symbolwith the result shifted up to values of +1 or −1 by multiplying by 2 andsubtracting 1.

In the present embodiment of the invention, the pre-modulation chipsequence α_(k) is generated such that the Laurent PAM expansion resultsin antipodal signaling for the main PAM pulse, h₀(t). Since the main PAMpulse represents most, but not all, of the energy in the baseband CPMsignal, the resulting spread spectrum GMSK signal is antipodal in ageneral sense. That is, the resulting spread spectrum GMSK signal may beantipodal to an extent that satisfies performance requirements, but itmay not be perfectly antipodal. Thus, for each data symbol, if the datasymbol takes on a first binary value, a corresponding portion of thespread spectrum GMSK signal exhibits a select waveform, and if the datasymbol takes on a second binary value, the corresponding portion of thespread spectrum GMSK signal exhibits a phase-inverted version of theselect waveform. The phase-inverted version of the select waveform maybe a nearly exact, but not perfectly exact, phase-inverted replica ofthe select waveform.

Demodulator Structure

FIG. 4 is a basic block diagram of a demodulator structure 110 forreceiving a spread spectrum GMSK signal, in accordance with oneembodiment of the present invention. For antipodal signaling, an optimumsymbol-by-symbol demodulator correlates the received signal with theexpected signal under the hypothesis, d_(n)=1, and compares the realpart of the output to zero (after phase rotation to remove the phaseerror). The correlator output is for the n^(th) symbol is,Z _(n) =∫r(t)[x _(n)(t)+jy _(n)(t)]*dt  (10)

The signal r(t) is the received complex valued (I and Q components)baseband signal, x(t) is the real part of the expected signal withd_(n)=1 and y(t) is the imaginary part of the expected signal. Theintegration is performed over to time duration, which is occupied by onesymbol of K chips. The simple demodulator structure 110 can be derivedby assuming all the PAM pulses to be negligible except the main pulseh_(o)(t). By using this assumption, the expected signal over the n^(th)symbol interval can be well approximated by

$\begin{matrix}\begin{matrix}{{{x_{n}(t)} + {{jy}_{n}(t)}} = {\sum\limits_{k = 0}^{K - 1}{a_{0,k}{h_{0}\left( {t - {kT}_{c}} \right)}}}} \\{= {{\underset{k^{\prime} = {odd}}{\sum\limits_{k^{\prime} = 0}^{K - 1}}{x_{{nK} + k^{\prime}}{h_{0}\left( {t - {\left( {{nK} + k^{\prime}} \right)T_{c}}} \right)}}} +}} \\{j{\underset{k^{\prime} = {even}}{\sum\limits_{k^{\prime} = 0}^{K - 1}}{y_{{nK} + k^{\prime}}{h_{0}\left( {t - {\left( {{nK} + k^{\prime}} \right)T_{c}}} \right)}}}} \\{x_{k} = {{Re}\left\{ a_{0,k} \right\}}} \\{y_{k} = {{Im}\left\{ a_{0,k} \right\}}}\end{matrix} & (11)\end{matrix}$thus the correlation in equation (10) becomes,

$\begin{matrix}{Z_{n} = {{{\sum\limits_{k^{\prime} = {odd}}^{\;}{x_{k}r_{k}}} - {j{\sum\limits_{k^{\prime} = {even}}^{\;}{y_{k}r_{k}\mspace{20mu} k}}}} = {{nK} + k^{\prime}}}} & (12)\end{matrix}$where r_(k) is the output (sampled once per chip) of a filter matched tothe main Laurent pulse,r _(k) =∫r(t)h _(o)(t−kT _(c))  (13)

Referring to FIG. 4, received signal(s) are provided to one or morefilters 112 matched to the main Laurent pulse h₀(t). The sampledfiltered output r_(k) is provided to a de-multiplexer block 114, whichoutputs separated even and odd samples of r_(k). The even and oddsamples of r_(k) are provided to correlator blocks 116. A PN sequenceblock 118 generates the same spread spectrum code b_(k) originally usedby the transmitter to generate the spread spectrum GMSK signal. Thespread spectrum code b_(k) is provided to a de-multiplexer 120, whichoutputs separated even samples y_(k) and odd samples x_(k). The even andodd samples y_(k) and x_(k) are also provided to the correlator blocks116.

In this manner, correlator blocks 116 correlate the received signal withthe expected signal under the hypothesis d_(n)=1. This results incorrelation outputs Z_(even) and Z_(odd), which are combined at acombiner block 122 to produce a complex result Z. The complex result Zis provided to a phase rotation block 124, which performs a phaserotation operation to compensate for any phase error. This results in anoutput Z_(rot). The phase-rotated output Z_(rot) may then be used toestimate the data symbols carried in the spread spectrum GMSK signal.

When timing and phase is not known, such as during the preamble (burstacquisition), the sample rate out of the chip matched filter, h_(o)(t),may be increased to allow several different timing hypothesis to beattempted. A sample rate of 2R_(c) may enable 4 timing hypothesis permain Laurent pulse (one I or Q channel pulse per two chips). Inaddition, a non-coherent metric, such as |Z|², may be used toaccommodate the unknown, random phase.

Performance

For large channel bit error rates (BERs), the channel BER of the spreadspectrum GMSK waveform can be accurately approximated (details omitted)by,

$\begin{matrix}{P_{b} = {Q\left( \sqrt{\frac{2\frac{E_{s}}{N_{o}}{H_{0}(0)}}{1 + {4\frac{E_{s}}{N_{o}}\frac{\alpha}{K^{2}{H_{0}(0)}}} + {2\frac{E_{s}}{N_{0}}\frac{\beta}{K \cdot {H_{0}(0)}}}}} \right)}} & (14)\end{matrix}$

Comparing equation (14) to the standard BER expression for coherentBPSK, one can see that degradation occurs due to the term H₀(0) in thenumerator and the second two terms in the denominator. The term H₀(0) isthe energy in the first Laurent Pulse, h₀(t). This value is slightlyless than unity. The second term in the denominator is due to theinterference with the h_(o)(t) pulses from the chips in the two adjacentsymbols. The third term is the interference with the higher orderLaurent pulses from the chips in the current symbol as well as theadjacent symbols. The terms α and β are constants which are functions ofthe cross correlation of the Laurent pulses.

By examining equation (14), the interference terms in the denominatorget smaller as K gets larger and get larger as the interferer signal tonoise ration (Es/No) gets larger. Thus it is expected that thedegradation to be larger at higher Es/No and to be independent of Es/Noas K gets large. For large values of K the performance approaches thatof theoretical coherent BPSK.

FIG. 5 is a plot of computer simulation results for channel BERperformance versus Es/No. The simulations were performed for a BTproduct of 0.10. These results confirm the analytical observationsregarding equation (14). As expected, the degradation becomes fixed forall values of Es/No as the number of chips per symbol (K) increases to24. At this point, the second two terms in the denominator arenegligible. In fact, at the Es/No range of interest (−4 dB≦Es/No≦0 dB),the inter-chip interference terms are negligible even for the minimumvalue of K (4 chips per symbol). Therefore, the degradation with respectto theoretical coherent BPSK in the interesting range of Es/No can beaccurately approximated as just the energy in the first Laurent pulse,h_(o)(t). This approximation is expected to hold for all values of K.FIG. 6 is a plot of the calculated loss, with respect to theoreticalcoherent BPSK, versus different values of the GMSK BT product.

Spread Spectrum GMSK Spectral Characteristics

Another dimension to the selection of the BT product is the spectraloccupancy. FIG. 7 illustrates the calculated power spectral density(PSK) of various modulated signals, including offset QPSK, QPSK, MSK,and GMSK signals of different BT products. These results are calculatedusing analytical techniques. Ideally, the spectrum of the transmittedsignal would have a rectangular “boxcar” shape. This would allow one toplace all of the signal energy in the allocated bandwidth and noneoutside of the allocated bandwidth. No known constant envelope signalhas this characteristic. As demonstrated in the figure, the GMSK signalfamily possesses roll-off which is much faster than that other constantenvelop waveforms considered.

Spectral characteristics can be further improved by using several FDMAchannels, each of smaller chip rate and consisting of, for example manyCode Division Multiple Access (CDMA) waveforms, to support the networktraffic instead one single channel of a larger chip rate. An examplewill illustrate this best. Suppose one has purchased 30 MHz of bandwidthand it is required that the PSD outside of the purchased bandwidth be 30dB down. First consider the use of a single spread spectrum GMSK signalusing a BT product of 0.25. When a BT product of 0.25 is used, the 30 dBbandwidth (double sided) of the signal is about 1.3 times the chip rate.The maximum chip rate that could be supported is 23.08 MHz.

Now consider the use of N FDMA channels in the 30 MHz purchasedbandwidth. The signals in each channel uses spread spectrum GMSKmodulation with a BT product of 0.25. The spacing between channels isset to be 0.75 times the chip rate. At this spacing the PSD's ofadjacent channels cross at the −10 dB point. This is conservative and inall likelihood, a smaller spacing between channels could be used. Thecomposite spectrum of N channels will have a 30 dB bandwidth of,W=(N−1)ΔR _(c) +BW ₃₀ R _(c)   (15)where W is in Hz, ΔR_(c), is the channel spacing, and BW₃₀ is the 30 dBbandwidth of a single GMSK signal normalized by the chip rate. Supposethere are N=10 channels. The spacing Δ is 0.75, and BW₃₀=1.3. Thus the30 dB bandwidth of sum of all signals is 8.05 time the chip rate. Themaximum chip rate for each signal will be 3.73 MHz (=30 MHz/8.05). Sincethere are 10 FDMA channels, the effective chip rate for this system is37.27 MHz, a factor of 1.6 greater than the example with 1 FDMA channel.

As seen in the example above, one can gain more physical layer capacityby using several FDMA channels each of smaller chip rate. The gain isdue to the resulting composite spectrum looking more like the ideal“boxcar” spectrum. The more FDMA channels used, the more ideal thespectrum will look. However, there is a diminishing return on increasingN, so the question is how many big should N be. Using a normalizedchannel spacing (normalized by the chip rate) of Δ and a normalized 30dB bandwidth of BW₃₀, the maximum chip rate that can be supported by asingle FDMA channel approach is W/BW₃₀ where W is the allocatedbandwidth. Using N FDMA carriers, the maximum chip rate that can besupported by each carrier is R_(c)=W/[(N−1)Δ+BW₃₀]. The effective chiprate, due to all N channels, will be the N times as large. The ratio ofthe multi channel effective chip rate to the single channel chip ratedetermines the capacity gain (as previously shown, the capacity of aCDMA system is proportional to the chip rate)

$\begin{matrix}{{Gain} = \left\lbrack {{\left( \frac{N - 1}{N} \right)\frac{\Delta}{{BW}_{30}}} + \frac{1}{N}} \right\rbrack^{- 1}} & (16)\end{matrix}$

As N becomes large, the gain goes to BW₃₀/Δ. This represents the maximumpossible gain. Selection of N=5 will yield more than 85% of the maximumachievable gain. Selection of N=10 will more than 90% of the maximumachievable gain.

Multiple Access Interference (MAI)

The final dimension in the selection of BT product is the performance ofthe spread spectrum GMSK signal in the presence of MAI. GMSK withdifferent values of BT will have different BER performance to the samelevels of MAI. Intuitively, one would expect that use of GMSK withhigher value BT would provide more resistance to MAI than lower of BTvalues since the occupied bandwidth is larger for the same chip rate.This is indeed the case, and the relationship has been quantified usingthe approach described below.

For N, asynchronous MAI interfering sources (all of the same powerlevel), the classical equation for relating N to Eb/No is

$\begin{matrix}{\frac{E_{b}}{I_{o}} = {\frac{W/R_{b}}{N} = \frac{\beta\;{R_{c}/R_{b}}}{N}}} & (17)\end{matrix}$

The only ambiguity is the definition of the bandwidth W. The bandwidthmay be defined as W=βR_(c). The constant β is a function of the BTproduct. A larger value of β results in more MAI resistance since theresulting Eb/Io is larger. The value of β is defined such that thevariance of the MAI out of the GMSK matched filters is identical to thevariance of the thermal noise out of the GMSK matched filters when theEb/No is set to a level of βR_(c)/(R_(b)N). This allows the totalEb/(No+Io) to be computed as the parallel combination of E_(b)/N_(o) andE_(b)/I_(o).

As an example, suppose that the variance of the MAI out of the GMSKmatched filter is measured to be σ² when R_(c)/R_(b) is 100 and N is 50.The variance out of the GMSK matched filters is also measured to be σ²when the Eb/No is set to 2.5 dB. Thus the performance with the above MAIand no thermal noise scenario is identical (in terms of the second orderstatistics out of the matched filter) to that of the thermal noise onlyat an E_(b)/N_(o) of 2.5 dB. to aim is to select a constant β such thatthe resulting E_(b)/I_(o) is 2.5 dB. This results in value of β=0.8891.FIG. 8 is a plot showing the constant β tabulated for different valuesof the BT product.

Optimal Selection of BT Product

FIG. 9 is a showing values of a figure of merit (FOM) for variousbandwidth definitions and BT products. Here, the occupied bandwidth of asingle GMSK signal is W=αR_(c), where the definition of occupiedbandwidth corresponds to some PSD bandwidth, such as 10 dB bandwidth or30 dB bandwidth. As previously shown, the CDMA capacity is proportionalto β/α. Thus, a good FOM for optimizing the BT selection is β/α, whichis illustrated in the table. A larger FOM indicates a better waveformchoice. For comparison purposes, regular MSK is also shown. Thebandwidth definition is the ratio of the PSD at the edge of band to thePSD at band center.

When it is required for the PSD of the signal to be very low out ofband, like 40 or 60 dB lower than center of band, a small value of theBT product of about 0.2 will optimize the capacity. When a larger out ofband PSD can be tolerated, a higher value of BT, or even MSK, should beused. For a 30 dB out of band PSD, the optimum value of BT is about 0.5.For certain multi-channel applications, carriers will be spaced as closetogether as possible. The bandwidth of interest for these carriers ismore like a −10 dB, or greater bandwidth. Thus one would think thatplain MSK might be the best modulation choice. However, it may bedesired that the PSD of the two edge channels roll off faster since theywill determine the bandwidth occupancy of the composite return channel(all FDMA channels). Regulatory or other constraints may dictate, forexample, that the relevant bandwidth definition for the two edgechannels be around 30 db, which indicates an optimum selection ofBT=0.5. By looking at the table, it is clear that there is very littledegradation in the FOM by using a BT product of 0.5 when the PSD rolloff of interest is in the −6 to −20 dB range. In practice, one mightwant to use a BT product slightly smaller than 0.5 to compensate forsome small amount spectral expansion in the modulator implementation.Thus, in one embodiment of the present invention, the optimum selectionof BT product will generally lie in the 0.4 to 0.5 range.

While the present invention has been described in terms of specificembodiments, it should be apparent to those skilled in the art that thescope of the present invention is not limited to the described specificembodiments. The specification and drawings are, accordingly, to beregarded in an illustrative rather than a restrictive sense. It will,however, be evident that additions, subtractions, substitutions, andother modifications may be made without departing from the broaderspirit and scope of the invention as set forth in the claims.

1. A method for transmitting a modulated signal comprising: obtaining asequence of data symbols; obtaining a spread spectrum code comprising asequence of spread spectrum chips; generating a sequence ofpre-modulation chips by combining the sequence of data symbols with thespread spectrum chips, wherein for each data symbol, at least one of thepre-modulation chips is generated by taking into account at least thedata symbol and at least one of the spread spectrum chips; performingGaussian Minimum Shift Keying (GMSK) modulation using the sequence ofpre-modulation chips to produce a spread spectrum GMSK signal;transmitting the spread spectrum GMSK signal; wherein the spreadspectrum GMSK signal adopts antipodal signaling to represent each datasymbol; and wherein for each data symbol, if the data symbol takes on afirst binary value, a corresponding portion of the spread spectrum GMSKsignal exhibits a select waveform, and if the data symbol takes on asecond binary value, the corresponding portion of the spread spectrumGMSK signal exhibits a phase-inverted version of the select waveform. 2.The method of claim 1 wherein the spread spectrum GMSK signal is capableof being approximated as a pulse amplitude modulation (PAM) signal. 3.The method of claim 2 wherein even ones and odd ones of the sequence ofpre-modulation chips separately correspond to in-phase (I) andquadrature (Q) components of the PAM signal.
 4. The method of claim 1wherein the spread spectrum code is a pseudo-noise (PN) code.
 5. Themethod of claim 1 wherein the GMSK modulation adopts a BT product withina range from 0.4 to 0.5.
 6. A method for transmitting a modulated signalcomprising: obtaining a sequence of data symbols; obtaining a spreadspectrum code comprising a sequence of spread spectrum chips; generatinga sequence of pre-modulation chips by combining the sequence of datasymbols with the spread spectrum chips, wherein for each data symbol, atleast one of the pre-modulation chips is generated by taking intoaccount at least the data symbol and at least one of the spread spectrumchips; performing Gaussian Minimum Shift Keying (GMSK) modulation usingthe sequence of pre-modulation chips to produce a spread spectrum GMSKsignal; and transmitting the spread spectrum GMSK signal; wherein eachdata symbol corresponds to K spread spectrum chips and K pre-modulationchips, wherein K is a positive integer; wherein a leading one of the Kpre-modulation symbols corresponding to a current data symbol isgenerated by taking into account i. an immediately preceding datasymbol; ii. a final one of the K spread spectrum chips corresponding tothe immediately preceding data symbol; iii. the current data symbol; andiv. a leading one of the K spread spectrum chips corresponding to thecurrent data symbol; and wherein remaining ones of the K pre-modulationsymbols corresponding to the current data symbol are each generated bytaking into account i. an immediately preceding one of the K spreadspectrum chips corresponding to the current data symbol; and ii. acurrent one of the K spread spectrum chips corresponding to the currentdata symbol.
 7. The method of claim 6 wherein even ones and odd ones ofthe remaining pre-modulation symbols corresponding to the current datasymbol are generated using opposite signs.
 8. An apparatus fortransmitting a modulated signal comprising: a data processing unitcapable of receiving a sequence of data symbols and combining thesequence of data symbols with a spread spectrum code comprising asequence of spread spectrum chips to produce a sequence ofpre-modulation chips, wherein for each data symbol, at least one of thepre-modulation chips is generated by taking into account at least thedata symbol and at least one of the spread spectrum chips; a modulatorcoupled to the data processing unit and configured to perform GaussianMinimum Shift Keying (GMSK) modulation on the sequence of pre-modulationchips to produce a spread spectrum GMSK signal; wherein the spreadspectrum GMSK signal adopts antipodal signaling to represent each datasymbol; and wherein for each data symbol, if the data symbol takes on afirst binary value, a corresponding portion of the spread spectrum GMSKsignal exhibits a select waveform, and if the data symbol takes on asecond binary value, the corresponding portion of the spread spectrumGMSK signal exhibits a phase-inverted version of the select waveform. 9.The apparatus of claim 8 wherein the spread spectrum GMSK signal iscapable of being approximated as a pulse amplitude modulation (PAM)signal.
 10. The apparatus of claim 9 wherein even ones and odd ones ofthe sequence of pre-modulation chips separately correspond to in-phase(I) and quadrature (Q) components of the PAM signal.
 11. The apparatusof claim 8 wherein the spread spectrum code is a pseudo-noise (PN) code.12. The apparatus of claim 8 wherein the GMSK modulation adopts a BTproduct within a range from 0.4 to 0.5.
 13. An apparatus fortransmitting a modulated signal comprising: a data processing unitcapable of receiving a sequence of data symbols and combining thesequence of data symbols with a spread spectrum code comprising asequence of spread spectrum chips to produce a sequence ofpre-modulation chips, wherein for each data symbol, at least one of thepre-modulation chips is generated by taking into account at least thedata symbol and at least one of the spread spectrum chips; a modulatorcoupled to the data processing unit and configured to perform GaussianMinimum Shift Keying (GMSK) modulation on the sequence of pre-modulationchips to produce a spread spectrum GMSK signal; wherein each data symbolcorresponds to K spread spectrum chips and K pre-modulation chips,wherein K is a positive integer; wherein a leading one of the Kpre-modulation chips corresponding to a current data symbol is generatedby taking into account i. an immediately preceding data symbol; ii. afinal one of the K spread spectrum chips corresponding to theimmediately preceding data symbol; iii. the current data symbol; and iv.a leading one of the K spread spectrum chips corresponding to thecurrent data symbol; and wherein remaining ones of the K pre-modulationchips corresponding to the current data symbol are each generated bytaking into account i. an immediately preceding one of the K spreadspectrum chips corresponding to the current data symbol; and ii. acurrent one of the K spread spectrum chips corresponding to the currentdata symbol.
 14. The apparatus of claim 13 wherein even ones and oddones of the remaining pre-modulation chips corresponding to the currentdata symbol are generated using opposite signs.
 15. A system fortransmitting a modulated signal comprising: means for obtaining asequence of data symbols; means for obtaining a spread spectrum codecomprising a sequence of spread spectrum chips; means for generating asequence of pre-modulation chips by combining the sequence of datasymbols with the spread spectrum chips, wherein for each data symbol, atleast one of the pre-modulation chips is generated by taking intoaccount at least the data symbol and at least one of the spread spectrumchips; means for performing Gaussian Minimum Shift Keying (GMSK)modulation using the sequence of pre-modulation chips to produce aspread spectrum GMSK signal; means for transmitting the spread spectrumGMSK signal; wherein the spread spectrum GMSK signal adopts antipodalsignaling to represent each data symbol; and wherein for each datasymbol, if the data symbol takes on a first binary value, acorresponding portion of the spread spectrum GMSK signal exhibits aselect waveform, and if the data symbol takes on a second binary value,the corresponding portion of the spread spectrum GMSK signal exhibits aphase-inverted version of the select waveform.